Abstract: In this paper we study a nonlinear eigenvalue problem driven by the -Laplacian. Assuming for the right-hand side nonlinearity only unilateral and sign conditions near zero, we prove the existence of three nontrivial solutions, two of which have constant sign (one is strictly positive and the other is strictly negative), while the third one belongs to the order interval formed by the two opposite constant sign solutions. The approach relies on a combination of variational and minimization methods coupled with the construction of upper-lower solutions. The framework of the paper incorporates problems with concave-convex nonlinearities.

10.Paul
H. Rabinowitz, Minimax methods in critical point theory with
applications to differential equations, CBMS Regional Conference
Series in Mathematics, vol. 65, Published for the Conference Board of
the Mathematical Sciences, Washington, DC; by the American Mathematical
Society, Providence, RI, 1986. MR
845785